This method of keeping the level book is adopted by many surveyors; the accuracy of the castings is proved by taking the difference of the sum of the back sights and the sum of the last fore sights, leaving out all the intermediate sights, and the difference of the last reduced level, and the same brought forward, and when these differences agree, the castings are correct. Some put the intermediate sights in a separate column, in this case a somewhat different method of casting is adopted :-I have given this method at page 357 of my system of levelling in the ninth edition of Nesbitt's Surveying.

B. M. foot of post.
Top of said post.

PLOTTING THE WORKING SECTION.

(See Plate III.)

Having drawn the datum line, prick off every chain and number them beneath the datum line to prevent mistakes. Next prick off the distances in the column of distances, and erect the perpendiculars, and just above the datum line, the height at each chain's length should be inserted, from the column of reduced levels; some also insert the heights of the intermediate stations, that the section may be plotted without the fear of errors. The horizontal scale in the example is 1 inch to 5 chains, and the vertical scale 1 inch to 25 feet; these scales are frequently adopted in parliamentary plans. Having laid off all the heights on the perpendiculars, the undulating line forming the surface of the section may be readily drawn, as shewn in the plate referred to, and the description of objects worthy of notice must be added. The section is then prepared for putting on the gradients, &c.

THE METHOD OF LAYING OUT GRADIENTS.

Gradients are evenly ascending or descending portions of a railway; the ascent or descent being always less than 1 foot, reckoned vertically, to 100 feet on the level, or estimated horizontally; thus to afford the means of rapid locomotive traction. Some gradients, however, are level, and such gradients are preferable to any other; but the undulations of the earth's surface prevent their adoption to any great extent, in by far the greater portion of railways.

The extreme left hand point of the section M, Plate III., being the point of junction with another railway, is the place from whence the gradients are to be laid out: the railway is represented by two parallel lines, the upper one being the surface of the rails, and the lower one the bottom of the ballasting or formation level, being 21 feet below the surface of the rails. For a short distance M N the gradient is level; the gradient NO then rises at the rate of 20 feet per mile, or one foot in 264, for the twofold object of diminishing the great cutting and getting sufficiently high over the road at stake 1064 to allow headway for public carriages to pass under the railway. From this point the gradient OP falls with the same rate of inclination for a considerable distance, the object being to get as low down as convenient further to the eastward where there would be a considerable embankment required; thus reducing the extent of both the cuttings and embankments. Each change of gradient is marked by a strong vertical line from the datum

line to the point of change, and the height written thereon The quantities of earthwork to form the cuttings and embankments with different slopes should be written on them, as shewn in the example (the method of finding these quantities shall be hereafter shewn); also above the line of figures denoting the height of the surface above the datum, should be placed the depth of the cutting from the surface to formation level at the same point, or the height of the enbankment, as the case may be: these heights and depths are those from which the calculations of the quantities of earthwork are to be made, and there fore must be strictly correct: these heights and depths are fre quently taken by measuring them carefully with the vertica. scale of the section, but they may be more correctly obtained by calculation in the following manner. Let it be required to find the depth of the cutting at stake No. 1083, when the height of the surface above the datum is 344.78 feet; at stake No. 1064 the height of formation level above datum is 269-20, from which point the gradient descends at the rate of 20 feet per mile, or 0.25 feet per chain, towards No. 1083; the distance from 1064 to 1083 is 19 chains, which multiplied by 0.25 gives 4.75 for the fall of the railway in the space between the two points; hence the height of the railway above datum at No. 1083 is 269-20 - 4.75 264.45; this sum subtracted from the whole height of the surface gives 344.78 - 264.45 = 80.33 for the depth of the cutting at that point, and so on for all the remaining numbers. In the same manner the heights of bridges and viaducts may be found, either below or above the railway, recollecting to subtract the reduced level from the height of the gradient in the former case.

It will here be proper to inform the student that the common, and, perhaps only practical method of laying out the gradients of railways, is by applying one end of an extended silken thread to the section at the commencement of the railway, the other end being so applied that the thread may cut the profile of the earth's surface, so as to leave equal portions of space both above and below the thread, judging by the eye, that the cuttings from the parts above the thread may furnish sufficient materials to fill up the spaces or parts below the thread to form the embankments. If the position of the first gradient, though in itself favourable, should cause the next gradient to be less favourable, with regard to the extent of cuttings and enbankments, the position of the first gradient must be altered to suit the next following gradient, till it be found that the compound result of the cuttings and embankments on the two gradients,